NDA & NA Exam (II)- 2017 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2017 - Mathematics

NDA & NA Exam (II)- 2017 - Mathematics Defence Questions

21.
The smallest positive integer n for which \( \left(\frac{1+i}{1-i}\right)^{n} = 1 \) is
A.
1
B.
4
C.
8
D.
16
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ANSWER :
B. 4
22.
If we define a relation R on the set \( N \times N \) as \( (a, b) R (c, d) \Leftrightarrow a + d = b + c \) for all \( (a, b), (c, d) \in N \times N \), then the relation is
A.
symmetric only
B.
symmetric and transitive only
C.
equivalence relation
D.
reflexive only
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ANSWER :
C. equivalence relation
23.
If \( y = x + x^{2} + x^{3} + \ldots \) up to infinite terms where \( |x| < 1 \), then which one of the following is correct?
A.
\( x = \frac{y}{1+y} \)
B.
\( x = \frac{y}{1-y} \)
C.
\( x = \frac{1+y}{y} \)
D.
\( x = \frac{1-y}{y} \)
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ANSWER :
A. \( x = \frac{y}{1+y} \)
24.
If \( \alpha \) and \( \beta \) are the roots of the equation \( 3x^{2} + 2x + 1 = 0 \), then the equation whose roots are \( \alpha + \beta^{-1} \) and \( \beta + \alpha^{-1} \) is
A.
\( 3x^{2} + 8x + 16 = 0 \)
B.
\( 3x^{2} - 8x - 16 = 0 \)
C.
\( 3x^{2} + 8x - 16 = 0 \)
D.
\( x^{2} + 8x + 16 = 0 \)
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ANSWER :
A. \( 3x^{2} + 8x + 16 = 0 \)
25.
The value of \( \frac{1}{\log_{3} e} + \frac{1}{\log_{3} e^{2}} + \frac{1}{\log_{3} e^{4}} + \ldots \) up to infinite terms is
A.
\( \log_{e} 9 \)
B.
0
C.
1
D.
\( \log_{e} 3 \)
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ANSWER :
A. \( \log_{e} 9 \)
26.
A tea party is arranged for 16 people along two sides of a long table with eight chairs on each side. Four particular men wish to sit on one particular side and two particular men on the other side. The number of ways they can be seated is
A.
\( 24 \times 8! \times 8! \)
B.
\( (8!)^{3} \)
C.
\( 210 \times 8! \times 8! \)
D.
16!
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ANSWER :
C. \( 210 \times 8! \times 8! \)
27.
The system of equations \( kx + y + z = 1 \), \( x + ky + z = k \) and \( x + y + kz = k^{2} \) has no solution if k equals
A.
0
B.
1
C.
−1
D.
−2
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ANSWER :
D. −2
28.
If \( 1 \cdot 3 + 2 \cdot 3^{2} + 3 \cdot 3^{3} + \ldots + n \cdot 3^{n} = \frac{(2n-1)3^{a} + b}{4} \), then a and b are respectively
A.
n, 2
B.
n, 3
C.
n + 1, 2
D.
n + 1, 3
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ANSWER :
D. n + 1, 3
29.
In \( \triangle PQR \), \( \angle R = \frac{\pi}{2} \). If \( \tan\left(\frac{P}{2}\right) \) and \( \tan\left(\frac{Q}{2}\right) \) are the roots of the equation \( ax^{2} + bx + c = 0 \), then which one of the following is correct?
A.
a = b + c
B.
b = c + a
C.
c = a + b
D.
b = c
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ANSWER :
C. c = a + b
30.
If \( \left|z - \frac{4}{z}\right| = 2 \), then the maximum value of |z| is equal to
A.
\( 1 + \sqrt{3} \)
B.
\( 1 + \sqrt{5} \)
C.
\( 1 - \sqrt{5} \)
D.
\( \sqrt{5} - 1 \)
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ANSWER :
B. \( 1 + \sqrt{5} \)
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